Project 4 -- alternative 1      CMSC 655   Nov. 13, 1996




     1. Read the excepts taken from the references [1,2]. This will
        fimiliarize you with the background, history, and meaning of
        the now famous "Lorentz equations".

     2. After having read the material, use one of the Matlab ODE 
        solvers (or write your own) to solve the following Lorentz
        system under various initial conditions and values of the
        parameter r and answer some questions related to the system.


                      =  -3(x - y),


                      =  -xz + rz - y,

                      
                      =   xy - z

        where r is a constant parameter.



        a) show that if r < 1 then the only critical point is

                  x = y = z = 0


           Does this seem to be a stable or unstable critical point?


        b) if r > 1 show that there is another critical point.


        c) Solve the above system numerically for r = 17 and initial
           conditions 
                   x(0) = -1, y(0) = -5.5, z(0) = 15,


           and
                I)  plot x(t) versus y(t) for 0 <= t <= 40.


                II) plot x(t),y(t),z(t) as a curve in three 
                    dimensional space for t between 0 and 40.



        d) Numerically solve the system for r = 26 and
            I) plot y(t)
              versus t for t going from 0 to 30 using the initial conditions
 
                             x(0) = y(0) = z(0) = 1 

            II) plot z(t) versus y(t) for the same range of t.


            III) plot y(t) versus x(t) for the same range of t.


            IV) get a three dimensional plot of the trajectory
                 (x(t),y(t),z(t))  as t varies from 10 to 30.


          

          references
          _________


          1, "Does God Play Dice? The Mathematics of Chaos", Ian Stewart
              Basil Blackwell


          2. "Nonlinear Waves, Solutions and Chaos", Eruk Enfeld and
              George Rowlands, Cambridge University Press